We compare the DFPV method to three competing methods, namely KPV [1]}, PMMR [1]}, and an autoencoder approach derived from the CEVAE method [3]}. In KPV, the bridge function is estimated through the two-stage regression as described in Section , where feature functions are fixed via their kernel functions. PMMR also models the bridge function using kernel functions, but parameters are learned by moment matching. CEVAE is not a PCL method, however it represents a state-of-the-art approach in correcting for hidden confounders using observed proxies. The causal graph for CEVAE is shown in Figure REF , and CEVAE uses a VAE [4]} to recover the distribution of confounder \(U\)  from the “proxy” \(Q\) . We make two modifications to CEVAE to apply it in our setting. First, we include both the treatment-inducing proxy \(Z\)  and output-inducing proxy \(W\)  as \(Q\)  in CEVAE (we emphasize that this does not follow the causal graph in Figure REF , since there exist arrows from \(Q\)  to \(A,Y\) ).
Second, CEVAE is originally used in the setting where \(Q\)  is conditioned on a particular value, whereas we marginalize \(Q\) .
See Appendix REF  for the choice of the network structure and hyper-parameters.
We tuned the regularizers \(\lambda _1,\lambda _2\)  as discussed in Appendix , with the data evenly split for Stage 1 and Stage 2. We varied the dataset size, and ran 20 simulations for each setting. Results are summarized in Figure REF .
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